A Bayesian framework to characterize ground motions even in the presence of missing data is developed. This approach features the combination of seismological knowledge (a priori knowledge) with empirical observations (even incomplete) via Bayesian inference. At its core is a Bayesian neural network model that probabilistically learns temporal patterns from ground motion data. Uncertainties are accounted for throughout the framework. Performance of the approach has been quantitatively demonstrated via various missing data scenarios. This framework provides a general solution to dealing with missing data in ground motion records by providing various forms of representation of ground motions in a probabilistic manner, allowing it to be adopted for numerous engineering and seismological applications. Notably, it is compatible with the versatile Monte Carlo simulation scheme, such that stochastic dynamic analyses are still achievable even with missing data. Furthermore, it serves as a complementary approach to current stochastic ground-motion models in data-scarce regions under the growing interests of PBEE (performance-based earthquake engineering), mitigating the data-model dependence dilemma due to the paucity of data, and ultimately, as a fundamental solution to the limited data problem in data scarce regions.